Laplace Transform And Its Applications

Made By:-S.Y. Electrical -2Shah Smit 96Sharma Pratik 98Shah Margeel 94Shaikh Rehan 97Shah Samarth 95 Laplace Transform And Its Applications

TopicsDefinition of Laplace TransformLinearity of the Laplace TransformLaplace Transform of some Elementary FunctionsFirst Shifting TheoremInverse Laplace TransformLaplace Transform of Derivatives & IntegralDifferentiation & Integration of Laplace TransformEvaluation of Integrals By Laplace TransformConvolution TheoremApplication to Differential EquationsLaplace Transform of Periodic FunctionsUnit Step FunctionSecond Shifting TheoremDirac Delta Function

Definition of Laplace TransformLet f(t) be a given function of t defined for all then the Laplace Transform ot f(t) denoted by L{f(t)} or or F(s) or is defined as

provided the integral exists,where s is a parameter real or complex.

Linearity of the Laplace TransformIf L{f(t)}= and then for any constants a and b

Laplace Transform of some Elementary Functions

First Shifting Theorem

Inverse Laplace Transform

Laplace Transform of Derivatives & Integral

Differentiation & Integration of Laplace Transform

Evaluation of Integrals By Laplace Transform

Convolution Theorem

Application to Differential Equations

Laplace Transform of Periodic Functions

Unit Step Function

Second Shifting Theorem

Dirac Delta function

Application of Laplace Transform

METHODOLOGY31Examples of nonlinear circuits:logic circuits, digital circuits,or any circuits where theoutput is not linearlyproportional to the input.

Examples of linear circuits:amplifiers, lots of OPMcircuits, circuits made ofpassive components (RLCs).If the circuit is a linear circuit

Laplace transform of the sourcesof excitation: s(t) S(s)Laplace transform of the all theelements in the circuitFind the output O(s) in theLaplace freq. domainObtain the time response O(t) bytaking the inverse LaplacetransformStop or approximatethe circuit into a linearcircuit and continue

NOYES

32KIRCHHOFFS VOLTAGE LAWConsider the KVL in time domain:

Apply the Laplace transform:

33OHMS LAWConsider the Ohms Law in time domain

Apply the Laplace transform

34INDUCTORInductors voltageIn the time domain:

In the s-domain:

35CAPACITORCapacitors currentIn the time domain:

In the s-domain:

36RLC VOLTAGEThe voltage across the RLC elements in the s-domain is the sum of a term proportional to its current I(s) and a term that depends on its initial condition.

Real-Life ApplicationsSemiconductor mobilityCall completion in wireless networksVehicle vibrations on compressed railsBehavior of magnetic and electric fields above the atmosphere

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